Theorem termco 49450

Description: The object of a terminal category. (Contributed by Zhi Wang, 17-Nov-2025.)

Hypotheses

Ref	Expression
termcbas.c	⊢ (𝜑 → 𝐶 ∈ TermCat)
termcbas.b	⊢ 𝐵 = (Base‘𝐶)

Assertion

Ref	Expression
termco	⊢ (𝜑 → ∪ 𝐵 ∈ 𝐵)

Proof of Theorem termco

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		termcbas.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ TermCat)
2		termcbas.b	. . 3 ⊢ 𝐵 = (Base‘𝐶)
3	1, 2	termcbas 49449	. 2 ⊢ (𝜑 → ∃𝑥 𝐵 = {𝑥})
4		unieq 4884	. . . . . 6 ⊢ (𝐵 = {𝑥} → ∪ 𝐵 = ∪ {𝑥})
5		unisnv 4893	. . . . . 6 ⊢ ∪ {𝑥} = 𝑥
6	4, 5	eqtrdi 2781	. . . . 5 ⊢ (𝐵 = {𝑥} → ∪ 𝐵 = 𝑥)
7		vsnid 4629	. . . . 5 ⊢ 𝑥 ∈ {𝑥}
8	6, 7	eqeltrdi 2837	. . . 4 ⊢ (𝐵 = {𝑥} → ∪ 𝐵 ∈ {𝑥})
9		id 22	. . . 4 ⊢ (𝐵 = {𝑥} → 𝐵 = {𝑥})
10	8, 9	eleqtrrd 2832	. . 3 ⊢ (𝐵 = {𝑥} → ∪ 𝐵 ∈ 𝐵)
11	10	exlimiv 1930	. 2 ⊢ (∃𝑥 𝐵 = {𝑥} → ∪ 𝐵 ∈ 𝐵)
12	3, 11	syl 17	1 ⊢ (𝜑 → ∪ 𝐵 ∈ 𝐵)

Syntax hints:   → wi 4   = wceq 1540  ∃wex 1779   ∈ wcel 2109  {csn 4591  ∪ cuni 4873  ‘cfv 6513  Basecbs 17185  TermCatctermc 49441

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-rab 3409  df-v 3452  df-dif 3919  df-un 3921  df-ss 3933  df-nul 4299  df-if 4491  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5110  df-iota 6466  df-fv 6521  df-termc 49442

This theorem is referenced by: (None)